Reverses of the Continuous Triangle Inequality for Bochner Integral of Vector-Valued Functions in Hilbert Spaces

Reverses of the Triangle Inequality in Banach Spaces

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.

New Reverses of Schwarz, Triangle and Bessel Inequalities in Inner Product Spaces

New reverses of the Schwarz, triangle and Bessel inequalities in inner product spaces are pointed out. These results complement the recent ones obtained by the author in an earlier paper. Further, they are employed to establish new Gruss type inequalities. Finally, some natural integral inequalities are stated as well.

Maximal and fractional maximal operators in the Lorentz Morrey spaces and their applications to the Bochner Riesz and Schrodinger type operators

The aim of this paper is to obtain boundedness conditions for the maximal function Mf and to prove the necessary and sufficient conditions for the fractional maximal oparator Ma in the Lorentz Morrey spaces which are a new class of functions. We get our main results by using the obtained sharp rearrangement estimates. The obtained results are applied to the boundedness of particular operators such as the Bochner Riesz operator and the Schrodinger type operators in the Lorentz Morrey spaces, where the nonnegative potential V belongs to the reverse Holder class.

Refinements of Reverse Triangle Inequalities in Inner Product Spaces

Refining some results of S. S. Dragomir, several new reverses of the triangle inequality in inner product spaces are obtained.

Quadratic Reverses of the Triangle Inequality in Inner Product Spaces

Some sharp quadratic reverses for the generalised triangle inequality in inner product spaces and applications are given.

Additive Reverses of the Generalised Triangle Inequality in Normed Spaces

Some additive reverses of the generalised triangle inequality in normed linear spaces are given. Applications for complex numbers are provided as well.

Some Reverses of the Generalised Triangle Inequality in Complex Inner Product Spaces

Some reverses for the generalised triangle inequality in complex inner product spaces that improve the classical Diaz-Metcalf results and applications are given.

More on Reverse Triangle Inequality in Inner Product Spaces

Refining some results of S. S. Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if $a$ is a unit vector in a real or complex inner product space $(H;< .,.>)$, $r, s>0, p\in(0,s], D=\{x\in H,\|rx-sa\|\leq p\}, x_1, x_2\in D-\{0\}$ and $ α_{r,s}=\min\{\frac{r^2\|x_k\|^2-p^2+s^2}{2rs\|x_k\|}: 1\leq k\leq 2 \}$, then $$\frac{\|x_1\|\|x_2\|-Re< x_1,x_2>}{(\|x_1\|+\|x_2\|)^2}\leq α_{r,s}.$$

Characterization of a generalized triangle inequality in normed spaces

For a normed linear space $(X,|\cdot|)$ and $p>0$ we characterize all $n$-tuples $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\|x_1+...+x_n\|^p\leq\frac{|x_1|^p}{μ_1}+...+\frac{|x_n|^p}{μ_n}$$ holds for any $x_1,...,x_n\in X$. We also characterize $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the reverse of the inequality above holds.

Reverse triangle inequality in Hilbert $C^*$-modules

We prove several versions of reverse triangle inequality in Hilbert $C^*$-modules. We show that if $e_1, ..., e_m$ are vectors in a Hilbert module ${\mathfrak X}$ over a $C^*$-algebra ${\mathfrak A}$ with unit 1 such that $<e_i,e_j>=0 (1\leq i\neq j \leq m)$ and $\|e_i\|=1 (1\leq i\leq m)$, and also $r_k,ρ_k\in\Bbb{R} (1\leq k\leq m)$ and $x_1, ..., x_n\in {\mathfrak X}$ satisfy $$0\leq r_k^2 \|x_j\|\leq {Re}< r_ke_k,x_j> ,\quad0\leq ρ_k^2 \|x_j\| \leq {Im}< ρ_ke_k,x_j> ,$$ then [\sum_{k=1}^m(r_k^2+ρ_k^2)]^{1/2}\sum_{j=1}^n \|x_j\|\leq\|\sum_{j=1}^nx_j\|, and the equality holds if and only if …

On Some Discrete Inequalities in Normed Linear Spaces

Some sharp discrete inequalities in normed linear spaces are obtained. New reverses of the generalised triangle inequality are also given.

量子均匀性测度的范数性质

Norm properties of quantum uniformity measures

该研究证明k=4时量子均匀性测度为范数，k≥5时三角不等式不成立，完整回答了Bu等人的开放问题，分别通过柯西-施瓦茨不等式与有限域线性方程组结构理论完成两类情形的证明。

非线性积分三角不等式的一个刻画

A characterization of a nonlinear integral triangle inequality

本文在 Banach 空间与 Lebesgue 测度空间上，刻画了使非线性积分三角不等式及其反向不等式成立的可测函数 u，并解决了离散对应问题。

A reverse Sidorenko inequality

Let $H$ be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph $G$ without isolated vertices, the weighted number of graph homomorphisms $\hom(G, H)$ satisfies the inequality \[ \hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)}, \] where $d_u$ denotes the degree of vertex $u$ in $G$. In particular, one has \[ \hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2} \] for every $d$-regular triangle-free $G$. The triangle-free hypothesis on $G$ is best possible. More generally, we prove a graphical Brascamp-Lieb type inequalit…

On inequalities for convex functions

We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the particular case where three aligned points are considered in a normed space.

Advances in Inequalities of the Schwarz, Gruss and Bessel Type in Inner Product Spaces

The main aim of this monograph is to survey some recent results obtained by the author related to reverses of the Schwarz, triangle and Bessel inequalities. Some Gruss' type inequalities for orthonormal families of vectors in real or complex inner product spaces are presented as well. Generalizations of the Boas-Bellman, Bombieri, Selberg, Heilbronn and Pecaric inequalities for finite sequences of vectors that are not necessarily orthogonal are also provided. Two extensions of the celebrated Ostrowski's inequalities for sequences or real numbers and the generalization of Wagner's inequality in…

Left m-invertibility by the adjoint of Drazin inverse and m-selfadjointness of Hilbert space operators

A Hilbert space operator $A\in\B$ is left $(X,m)$-invertible by $B\in\B$ (resp., $B\in\B$ is an $(X,m)$-adjoint of $A\in\B$) for some operator $X\in\B$ if $\triangle_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{m-j}XA^{m-j}=0$ (resp., $δ_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{(m-j)}XA^j=0$). No Drazin invertible operator $A\in\B$, with Drazin inverse $A_d$, can be left $(I,m)$-invertible (equivalently, $m$-invertible) by its adjoint or its Drazin inverse or the adjoint of its Drazin inverse. For Drazin inverrtible operat…

Sincov's inequalities on topological spaces

Assume that $X$ is a non-empty set and $T$ and $S$ are real or complex mappings defined on the product $X \times X$. Additive and multiplicative Sincov's equations are: $$T(x,z) = T(x, y ) + T(y, z)$$ and $$S(x,z) = S(x, y ) \cdot S(y, z),$$ respectively. Both equations play important roles in many areas of mathematics. In the present paper we study related inequalities. We deal with functional inequality $$ G(x,z) \leq G(x, y ) \cdot G(y, z), \quad x , y, z \in X $$ and we assume that $X$ is a topological space and $G\colon X \times X \to \mathbb{R}$ is a continuous mapping. In some our state…

Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón--Zygmund Operators

Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…

双曲Banach空间

Hyperbolic Banach spaces

本文提出双曲Banach空间的概念，通过替代传统椭圆框架中的线性组合、三角不等式和Cauchy完备性，探讨非光滑洛伦兹几何中的定向完备性，并研究L^p空间等结构在该框架下的应用。

Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces

Let $({\mathcal X},ρ,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, a…